If a company has a bulletin board at the workplace, does the employer have a right to restrict what can be posted on the bulletin board.
Because your other questions are clearly related to bulletin boards and unionization, I will focus on that issue here as well. In general, companies do have the right to restrict what is posted on their bulletin boards. The bulletin boards are their private property and they generally have the right to restrict the uses that are made of that private property. However, particularly with regard to unionization and other issues of worker rights, companies must be very careful.
First, companies need to have clear policies about what can and cannot be posted on their bulletin boards. A company cannot feel secure if it simply makes case-by-base decisions about what sorts of materials are acceptable. This leaves the firm open to claims that it is violating employee rights if it prevents employees from posting information about unions.
Second, companies have to have those policies in place before they are needed. A firm cannot simply see a posting that it does not like and then create a policy that bans that type of posting.
Finally, company rules about postings cannot be motivated by an anti-union bias. The rules have to have some purpose other than to prevent unionization.
Thus, employers do have the right to determine what goes up on their bulletin boards, but they have to make sure that they set up policies about the bulletin boards in the correct way.